Implementation of ultrasoft pseudopotentials in large-scale grid-based electronic structure calculations

An implementation of Vanderbilt ultrasoft pseudopotentials in real-space grid-based electronic structure calculations is presented. Efficient utilization of these pseudopotentials requires the use of different grids for (i) wave functions, (ii) charge density, and (iii) sharply peaked operators within the atomic core radii. High-order interpolation between the various grids is important for accuracy, as is high-order discretization for the differential operators. However, efficiency is also of paramount importance, especially when parallelizing over hundreds or thousands of processors. We describe algorithms and procedures used to achieve an effective implementation in the real-space multigrid code, and provide test results for first-row diatomics, bulk transition metals, and energy-conserving quantum molecular dynamics of water. The code parallelizes efficiently over several thousands of processors on modern parallel supercomputers, such as the Cray XT3 and XT4.

Figure 1

(Color online) Bond length in ${\mathrm{O}}_2$ molecule calculated with different B-splines orders. The bond length calculated with local cubic interpolation is $2.269\mathrm{a.u.}$

Figure 2

Equation of state for bulk gold in fcc structure. The continuous curve is the Murnaghan equation of state. The points are the calculated energies at the indicated volume.

Figure 3

(Color online) Band structure of bulk copper.

Figure 4

(Color online) Energies and temperature in molecular dynamics simulation of liquid water. In the upper plot, the dashed line represents the total energy. The ionic and Nosé-Hoover energies were shifted by $-156.035$ and $-155.945\mathrm{eV}$ in order to appear in one plot with other energies.

Figure 5

(Color online) Performance data for a system of 432 water molecules on Cray XT3. Both cores on each processor were used. Points represent data from calculations; the line is a guide for the eyes.

Figure 6

(Color online) Performance data for a system of 819 water molecules on Cray XT3. Points represent data from calculations; lines are guides for the eyes.